Instructional Shifts for Mathematics

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Instructional Shifts in Mathematics

1. Focus: Focus strongly where the Standards focus.

2. Coherence: Think across grades and link to major topics within grades.

3. Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application.

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Shift #1: Focus Strongly Where the Standards Focus

• Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom.

• Focus deeply on what is emphasized in the standards, so that students gain strong foundations.

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Focus

• Move away from "mile wide, inch deep" curricula identified in TIMSS.

• Learn from international comparisons.

• Teach less, learn more.

“Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.”

– Ginsburg et al., 2005

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Mathematics topics

intended at each grade by

at least two-thirds of A+

countries

Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states

The shape of math in A+ countries

1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).

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K 12

Number and Operations

Measurement and Geometry

Algebra and Functions

Statistics and Probability

Traditional U.S. Approach

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Focusing Attention Within Number and Operations

Operations and Algebraic Thinking

Expressions and Equations

Algebra

→ →

Number and Operations—Base Ten →

The Number System

→

Number and Operations—Fractions

→

K 1 2 3 4 5 6 7 8 High School

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Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2 Addition and subtraction - concepts, skills, and problem solving and place value

3–5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving

6 Ratios and proportional reasoning; early expressions and equations

7 Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra and linear functions

Key Areas of Focus in Mathematics

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Group Discussion

Shift #1: Focus strongly where the Standards focus.

In your groups, discuss ways to respond to the following question, “Why focus? There’s so much math that students could be learning, why limit them to just a few things?”

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Shift #2: Coherence: Think Across Grades, and Link to Major Topics Within Grades

• Carefully connect the learning within and across grades so that students can build new understanding on foundations built in previous years.

• Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

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Coherence: Think Across Grades

Example: Fractions

“The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.”

Final Report of the National Mathematics Advisory Panel (2008, p. 18)

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Group Discussion

Shift #2: Coherence: Think across grades, link to major topics within grades

In your groups, discuss what coherence in the math curriculum means to you. Be sure to address both elements—coherence within the grade and coherence across grades. Cite specific examples.

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Shift #3: Rigor: In Major Topics, Pursue Conceptual Understanding, Procedural Skill and Fluency, and Application

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Rigor

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• Requires a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving situations

• Pursuit of all three requires equal intensity in time, activities, and resources.

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Solid Conceptual Understanding

• Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives

• Students are able to see math as more than a set of mnemonics or discrete procedures

• Conceptual understanding supports the other aspects of rigor (fluency and application)

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Fluency

• The standards require speed and accuracy in calculation.

• Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts

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Application

• Students can use appropriate concepts and procedures for application even when not prompted to do so.

• Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS.

• Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content.

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Group Discussion

Shift #3: Rigor: Expect fluency, deep understanding, and application

In your groups, discuss ways to respond to one of the following comments: “These standards expect that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.”

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It Starts with Focus

• The current U.S. curriculum is "a mile wide and an inch deep."

• Focus is necessary in order to achieve the rigor set forth in the standards.

• Remember Hong Kong example: more in-depth mastery of a smaller set of things pays off.

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Cautions: This is...

• Not about “gap analysis”

• Not about buying a text series

• Not a march through the standards

• Not about breaking apart each standard

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Resources

www.achievethecore.org

www.illustrativemathematics.org

http://pta.org/parents/content.cfm?ItemNumber=2583&RDtoken=51120&userID

commoncoretools.me

www.corestandards.org

http://parcconline.org/parcc-content-frameworks

http://www.smarterbalanced.org/k-12-education/common-core-state-standards-tools-resources/

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